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You have free article s left. Already a subscriber? Sign in. See Subscription Options. Here are his comments in full: None of this is new—the same debate has been playing out for fifty years. Further Reading : Steven Pinker tipped me off to this splendidly scathing review in 3.

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With an endorsement like that, what more needs to be said? Indeed, this is a very interesting and very informative book, scarcely dimmed by the passage of years -- it was first published in ! In the interim, much has changed; one amusing example is the following statement on page "Today [] over corre On the cover is an interesting mini-review: "This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands.

In the interim, much has changed; one amusing example is the following statement on page "Today [] over correct decimals of the number pi are known. And yet much has not changed, and this book remains one of the best exposition of how our system of mathematics arose. One of the best chapters is the second chapter, where the author clearly describes how our modern Indo-Arabic numerical system, which is arguably the greatest mathematical discovery of all time, arose in India in the first few centuries of the common era, and from their percolated to the Arab world, and then to a kicking-and-screaming European world.

Dantzig introduces Chapter 2, where this is discussed, with this interesting quote from Laplace: "It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit.

But its very simplicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

Jun 24, Jane rated it liked it. The anthropological survey about number systems in the first few chapters was pretty interesting, but the dryness of the writing really came into the forefront when the later chapters turned to increasingly technical mathematics. While I appreciate the rigor, it ended up feeling like I was reading a math textbook, which is not my jam right now. Nov 10, Andi rated it really liked it Shelves: science , non-fiction , books-i-own , math. Loved this near closing line - The reality of today was but an illusion yesterday.

Apr 07, Dennis Littrell rated it it was amazing. Einstein called this "the most interesting book on the evolution of mathematics which has ever fallen into my hands. This is a republication of that fourth edition Dantzig died in edited by Joseph Mazur with a foreword by Barry Mazur.

Number: The Language of Science

It is an eminently readable book like something from the pages of that fascinating four-volume work The World of Mathematics edited by James R. Newman in that it is aimed a Einstein called this "the most interesting book on the evolution of mathematics which has ever fallen into my hands. Newman in that it is aimed at mathematicians and the educated lay public alike.

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Part history, part mathematics and part philosophy, Number is the story of how we humans got from "one, two Strange to say it is also about reality. Here is Dantzig's concluding statement from page in Appendix D: " Be it determinism or rationality, empiricism or the mathematical method, it has recognized that man is the measure of all things, and that there is no other measure. I was surprised to read them in a book on the theory of numbers, and even more surprised to realize that if mathematics is a distinctly human language, it is entirely possible that beings from distant worlds may speak an entirely different language; and therefore our attempts to use what many consider the "universal" language of mathematics to communicate with them may be in vain.

And this thought makes me wonder. Is the concept "two," for example, as opposed to the number "2" really just a human construction?

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Would not intelligent life anywhere be able to make a distinction, just as we have, between, say, two things and three things? And if so, would they not be able to count?

  • Number Sense in Humans | Fingerprints | InformIT.
  • Number: The Masterpiece Science Edition by Tobias Dantzig, Joseph Mazur | Waterstones.
  • Number - The Language Of Science, The Masterpiece Science Edition.

And would not then the entire edifice of mathematics or at least most of it follow? I wonder if Dantzig was not in contradiction with himself on this point because earlier he writes p. As to the numbers themselves putting philosophy aside we learn that the two biggest bugaboos in the history of number are zero and infinity. It took a long, long time for humans, as Dantzig relates, to accept the idea of zero as a number. Today zero is also a place-holder. But what does it mean to say that there are zero pink elephants dancing about my living room?

Giuseppe Longo - A Mathematical Critique of Computational Thinking in the Sciences of Nature

I can see one cow in the yard, or two or three, but I cannot see zero cows in the yard. Of course, today it is easy to see that zero is a number that is less than one and greater than minus one. I have one cow and I sell that one cow. Now I have zero cows. Curiously, note that the plural noun "cows" is grammatically required. However, the imperfect fit within the entire structure of mathematics that zero has achieved may be appreciated by realizing that every other number can be a denominator; that is, three over one equals three, three over two equals 1.

It is a convention of mathematics to say that division by zero is "undefined. I used to think when I was young that infinity was the proper answer to division by zero. For Dantzig this is clearly not correct because to him infinity is not a number at all but a part of the process. He writes, "the concept of infinity has been woven into the very fabric of our generalized number concept.

He notes in the next sentence that in the generalization to "real" numbers, "the limits of these processes" were "admitted The culmination of Dantzig's argument here is that infinity itself is a construction of the human mind and exists nowhere that we can prove outside of the human mind. He believes that the basis for our belief in the existence of infinity comes from our erroneous conception of time as a continuum. Dantzig notes that Planck time and indeed all aspects of the world are to be seen in terms of discrete quanta and not continuous streams.

Ultimately, Dantzig gives this sweeping advice to the scientist: " Apr 15, Anna rated it it was amazing Shelves: philosophy-read. This is another one I pick up a lot. There is some really dense math that is really outside my understanding, but also some incredibly lucid analysis of the development of mathematics and how it has effected the way we perceive and cognate.

Tremendous stuff, and humbling! Sep 01, Niharika rated it it was amazing. It is all our imagination. Mathematics is high art as it creates a whole new world like number system. In this book author changes your perception about numbers. The book goes from the history of creation of numbers in different societies to real, trancedental and complex numbers. Its a must read for anyone who is atall interested in mathematics. Jul 03, Jessica rated it it was ok. The first couple of chapters were interesting--about the evolution of counting and the development of language to describe abstract concepts like "how many", but after that, the book got extremely tedious and boring.

Not one of my favorites on math. Aug 04, Stephen Armstrong rated it it was amazing. Number theory clearly explained in this classic. Beautifully written. What book on number theory survives 77 years, unless it is extraordinary? Oct 23, Anthony Tenaglier rated it liked it.

Number. the Language of Science

This book was quite interesting to me; however, the latter portions were more technical than I could well tackle, given my limited mathematical expertise. I did not agree with the author's nominalist views on the ultimate nature of mathematics. My views are more "Platonist", given that I am convinced that mathematics has some form of objective existence, and in view of my belief that mathematics is discovered rather than created or invented. Tobias Dantzig believed that logic is a branch of math This book was quite interesting to me; however, the latter portions were more technical than I could well tackle, given my limited mathematical expertise.

Tobias Dantzig believed that logic is a branch of mathematics. I quote from page "How then can we arrive at a criterion [for the reality of the number concept]? Not by evidence, for the dice of evidence are loaded. Not by logic, for logic has no existence independent of mathematics. How then shall mathematical concepts be judged?

They shall not be judged! Mathematics is the supreme judge; from its decisions there is no appeal. Mathematics is a branch of logic. Since thinking that is non-mathematical can occur, it follows that logic is more fundamental than mathematics. Dantzig placed great emphasis on mathematical intuition as the ultimate guide for mathematics.

I believe that intuition is the faculty of the intellect that enables discovery of mathematics, but it is logic that is the arbiter of valid mathematical thought. Logic has the final word. Furthermore, logic is the principle behind all thought, but mathematics consists of logic APPLIED to quantifiable entities and structures. Thus, mathematics is subsumed in logic. I was also not impressed favorably by Dantzig's ridicule of Christianity. On page he wrote: "When, after a thousand-year stupor, European thought shook off the effect of the sleeping powders so skilfully administered by the Christian Fathers, the problem of infinity was one of the first to be revived.

Number. the Language of Science - AbeBooks - Tobias Dantzig:

Roman indifference and the long Dark Ages of religious obscurantism prevented a resumption of this process for fifteen hundred years. It was not in India, China, or the Muslim countries where science blossomed, but in the heart of Christian Europe. It is not at all obvious that this alleged "religious obscurantism" was, in fact, inhibiting mathematical and scientific advances.

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